This works as expected, when val is something like $1+x+x^2$. What I wanted to do is to calculate the value of $f$, but using a generic element of $GF(2^3)$, e.g. $a_2x^2+a_1x+a_0$. The idea is to have the result expressed in terms of the $a_i$. Is this possible in Sagemath?

I have tried to do it using symbolic variables, but they always belong to the Symbolic Ring, which (as far as I can tell) does not mix with other rings. Because this example is small, I was able to do the computations by hand; the value of having SAGE doing it is of course, to apply it to cases that are infeasible to do without a computer.

Comments

This is most helpful, but there is a detail still missing: what I wanted is a way for a0, a1 and a2 to be treated as generic elements of GF(2). Meaning for example, that $a_i^n=a_i$ and $a_i+a_i = 0$. Without this, the resulting expression is needlessly complicated. Can it be achieved?

I accepted this answer because, well, it fully answers my question. But now I must ask for what would truly be the cherry on top of the cake: is there any way to group the result by 'x'? I.e. to make the result of your example look like x*(a0*a1+a2) + a0*a1 + a2? I tried these methods, but they don't work. The closest I can get is sorted(r.monomials()), where r is the produced result. Again, thank you in advance.